The Banach Space C0

E extracta mathematicae Vol. 16, N´um.1, 1 – 25 (2001)

The Banach Space c0

Gilles Godefroy

Equipe d’Analyse, Universit´eParis VI, Case 186, 4, Place Jussien, 75252 Paris Cedex 05, France e-mail: [email protected]

(Survey paper presented by J.P. Moreno)

AMS Subject Class. (2000): 46B20 Received October 22, 2000

I. Introduction

We survey in these notes some recent progress on the understanding of the Banach space c0 an of its subspaces. We did not try to complete the (quite ambitious) task of writing a comprehensive survey. A few topics have been selected, in order to display the variety of techniques which are required in such investigations. It is hoped that this selection can provide the reader with some intuition about the behaviour of the space. Our choice has also been influenced by the open problems which conclude these notes. It is our hope that some of these problems are not desperately hard. The unit ball of c0 can be (vaguely) visualized as a cubic box with no vertices; somehow, only the internal parts of the sides remain. Cubic boxes are suitable for packing, and indeed it is easy to map a space to c0 (see § II). A cube is a polyhedron, and indeed the polyhedral nature of c0 and its subspaces inform us of some of their isometric properties (see § III). The space c0 contains “very few” compact sets, unlike spaces which do not contain c0 and retain some features of reflexive spaces (see § IV). The natural norm of c0 is flat at every point in the direction of a finite-codimensional space and c0 is the largest space with this property. This helps to prove that the unit ball of c0 is a sturdy box, whose shape cannot be altered by non-linear Lipschitz maps (see § V). Finally, vectors in c0 can easily be approximated by restricting their support to given finite subsets. Subspaces of c0 are in general not stable under this operation. However, smoothness properties of these subspaces sometimes help to build a decent approximation scheme (see § VI). A sample of open problems is displayed in the last section VII.

1 2 g. godefroy

Notation We denote by c0 the separable Banach space of all real sequences (un) such that limn→∞(un) = 0. This space is equipped with its canonical norm © ª k(un)k∞ = sup |un|; n ≥ 1 . Capital letters such as X, Y , ... will denote Banach spaces. By “sub- ∗ space”, we mean “closed subspace”. We denote by en the linear form deﬁned on c0 by ∗ en((uk)) = un. ∗ ∗ 1 Of course, en ∈ c0 = ` . We denote by NA(X) the subset of the dual space X∗ consisting of norm-attaining linear functionals. We otherwise follow the classical notation as it can be found e.g. in [32].

II. c0 is an easy target

It is quite easy to give a usable representation for operators T from a Banach space X to c0. Indeed, if we let

∗ ∗ ∗ ∗ T (en) = xn ∈ X then clearly ∗ T (x) = (xn(x))n≥1 (1) ∗ ∗ ∗ ∗ and the sequence (xn) is w -convergent to 0 in X . Conversely, if (xn) is a ∗ ∗ w -null sequence in X , then (1) deﬁnes an operator from X to c0. This very simple representation provides us with a bunch of operators from X into c0, at least when the space X is separable since in this case, bounded subsets of X∗ are w∗-metrizable and thus convergent sequences are easily constructed. This simple idea lies in the background of an old but important result on c0: Sobczyk’s theorem.

Theorem II.1. ([35]) Let X be a separable Banach space, and Y a subspace of X. Let T : Y → c0 be a bounded operator. There exists Te : X → c0 such that

e (i) T|Y = T . (ii) kTek ≤ 2kT k. the banach space c0 3

∗ ∗ ∗ ∗ Proof. ([36]) Let yn = T (en). Clearly kT k = supn kynk. By Hahn– ∗ ∗ ∗ ∗ ∗ Banach theorem, there exist xn ∈ X such that kxnk ≤ kT k and xn|Y = yn for all n ≥ 1. We set

K = {x∗ ∈ X∗; kx∗k ≤ kT k} and L = K ∩ Y ⊥. When equipped with the w∗-topology, K is metrizable compact. Let d be a distance which deﬁnes the weak-star topology on K. ∗ ∗ ∗ Since lim xn(y) = lim yn(y) = 0 for every y ∈ Y , every w -cluster point to ∗ the sequence (xn) belongs to L. It follows by compactness that the distance ∗ ∗ from (xn) to L in (K, d) tends to 0, hence we can pick tn ∈ L such that

∗ ∗ ∗ w - lim(xn − tn) = 0.

If we now deﬁne Te : X → c0 by ¡ ¢ e ∗ ∗ T (x) = (xn − tn)(x) n≥1 it is easily seen that Te works. Moreover ¡ ¢ e ∗ ∗ ∗ ∗ kT k = sup kxn − tnk ≤ sup kxnk + ktnk ≤ 2kT k. n n

Let us spell out two important applications.

Corollary II.2. Let X be a separable Banach space, and let Y be a subspace which is isomorphic to c0. Then Y is complemented in X.

Proof. Let T : Y → c0 be an isomorphism onto c0. In the notation of Theorem II.1, T −1 ◦ Te : X → Y is a projection.

Corollary II.3. Let X be a separable Banach space which has a subspace Y such that both spaces Y and X/Y are isomorphic to subspaces of c0. Then X is isomorphic to a subspace of c0.

Proof. Let T : Y → c0 and S : X/Y → c0 be isomorphisms onto subspaces of c0. We denote again Te : X → c0 an operator extending T , and Q : X → X/Y the canonical¡ map. It is easily¢ checked that V : X → c0 ⊕ c0 ' c0 deﬁned by V (x) = Te(x),S ◦ Q(x) is an isomorphism of X into c0. 4 g. godefroy

Remark II.4. Sobczyk’s theorem, and the above corollaries, fail in the nonseparable case. Indeed we can take X = `∞ in corollary II.2 and c0 is not complemented in `∞. There is a compact set K [9, Example VI.8.7.], such that C(K) = X contains a subspace Y isomorphic to c0(N) such that X/Y is isomorphic to c0(Γ), with |Γ| = c, but X is not a subspace of c0(Γ) since X is not weakly compactly generated. Although it is harder to map a nonseparable Banach space into a c0(Γ) space, such mappings play a fundamental role in the study of “nice” nonseparable spaces. We refer to the last three chapters of [9] for this topic.

III. Some isometric properties

It is easy to characterize spaces which are isometric to subspaces of c0 by applying the formula (1) from § II to an isometric embedding.

Proposition III.1. Let X be a Banach space. The following assertions are equivalent:

(i) X is isometric to a subspace of c0. ∗ ∗ ∗ ∗ (ii) There exists a sequence of (xn) in X such that w -limn(xn) = 0 and for all x ∈ X, © ∗ ª kxk = sup |xn(x)|; n ≥ 1 .

The proof follows immediately from the consideration of

¡ ∗ ¢ T (x) = xn(x) n≥1

∗ which is an isometric embedding into c0 if and only if (xn) satisfies (ii). This simple proposition has some interesting consequences. We first need a definition.

Definition III.2. Let X be a Banach space, and g : X → R a continuous function. We say that g locally depends upon ﬁnitely many coordinates if for ∗ every x ∈ X, there exist ε > 0, a ﬁnite subset {f1, f2, . . . , fn} of X and a n continuous function ϕ : R → R such that g(y) = ϕ(f1(y), f2(y), . . . , fn(y)) for every y such that kx − yk < ε.

When g : X → R is a norm, we say that it locally depends upon ﬁnitely many coordinates if the above condition holds for every x ∈ X \{0}. We note the banach space c0 5 that when this happens, ϕ can be chosen to be a norm on Rn. Indeed, let T : X → Rn be deﬁned by ¡ ¢ T (y) = fi(y) 1≤i≤n and pick any y ∈ SX with kx − yk < ε. For every u ∈ Ker T , the function

λ −→ ky + λuk is convex, and ≥ 1 on some neighbourhood of 0, hence ≥ 1 on R and we have

kykX/ Ker T = 1.

It follows that we can take ϕ(·) = k · kX/ Ker T .

Proposition III.3. Let X be a subspace of c0, and let k · k be the re- striction of k · k∞ to X. Then, for any x ∈ X \{0}, there are δ > 0 and a ∗ ∗ ﬁnite subset (xn)n∈J of X such that if kx − yk < δ, then

© ∗ ª kyk = sup |xn(y)|; n ∈ J . In particular, k · k depends locally of finitely many coordinates. Moreover, X is polyhedral; that is, the unit ball of every finite dimensional subspace of X has finitely many extreme points.

Proof. Pick x ∈ X \{0}. We have

∗ kxk = sup |en(x)| n

∗ and limn en(x) = 0. Hence if

∗ I = {n ≥ 1; |en(x)| < kxk} one has © ∗ ª kxk − sup |en(x)|; n ∈ I = ε > 0 and the set J = N \ I is ﬁnite. If y ∈ X is such that kx − yk < ε/3, one clearly has

For showing polyhedrality, we pick E ⊆ X a ﬁnite dimensional subspace. ∗ Let SE = {x ∈ E; kxk = 1}. For all x ∈ SE, there is ε(x) > 0 and F (x) ⊆ X a ﬁnite subset such that ky − xk < ε(x) implies © ª kyk = sup |x∗(y)|; x∗ ∈ F (x) .

Since SE is compact, there are x1, . . . , xn in SE such that

[n ¡ ¢ SE ⊆ B xi, ε(xi) . i=1

n ∗ If F = ∪i=1F (xi), F is a ﬁnite subset of X such that © ª kxk = sup |x∗(x)|; x∗ ∈ F

|F | for all x ∈ E, hence E is isometric to a subspace of (R , k · k∞) and polyhedrality follows.

Subspaces of c0 are typical examples of polyhedral spaces, but they are far from exhausting the class of polyhedral spaces. We refer to [13] for an up-to-date survey of this theory. We shall use these notions for investigating nearest points to subspaces in the class of subspaces of c0. We recall that a subspace Y of a Banach space X is called proximinal if for every x ∈ X, there exist y0 ∈ Y such that © ª kx − y0k = inf kx − yk; y ∈ Y .

If for instance Y = Ker(x∗) is an hyperplane, Y is proximinal if and only if x∗ ∈ NA(X), where NA(X) denotes the subset of X∗ consisting of norm- attaining functionals. That is, x∗ ∈ NA(X) when © ª sup |x∗(x)|; kxk ≤ 1 is attained. This observation has an easy useful generalization, namely:

Proposition III.4. ([15]) Let X be a Banach space, and Y a subspace of X of ﬁnite codimension. If Y is proximinal in X, then Y ⊥ is contained into NA(X). the banach space c0 7

Proof. Let Q : X → X/Y be the quotient map. Clearly Y is proximinal if and only if Q(BX ) = BX/Y . Since X/Y is ﬁnite dimensional, every linear form on (X/Y ) attains its norm. The result follows since Q∗ is an isometry from (X/Y )∗ onto Y ⊥. Proposition III.4 can be reformulated as follows: if Y ⊆ X is a proximinal subspace of ﬁnite codimension, every hyperplane H such that Y ⊆ H ⊆ X is proximinal. The converse does not hold in general [34]. We will show however that it holds in subspaces of c0. Indeed one has:

Theorem III.5. ([18]) Let X be a subspace of c0. Let Y be a ﬁnite codimensional subspace of X. Then Y is proximinal in X if and only if the orthogonal Y ⊥ of Y in X∗ is contained in NA(X).

Proof. We have to show that Y is proximinal if Y ⊥ is contained in NA(X). This is obtained by combining the following facts.

Lemma III.6. Let Y be a ﬁnite codimensional subspace of a Banach space X such that Y ⊥ is contained in NA(X) and Y ⊥ is polyhedral. Then Y is proximinal. Indeed X/Y = (Y ⊥)∗ is polyhedral as well and thus every extreme point e of B(Y ⊥)∗ is in fact exposed. That is, there is x∗ ∈ Y ⊥ with kx∗k = 1 and

{t ∈ (Y ⊥)∗; t(x∗) = 1} = {e}. (2)

But since x∗ ∈ NA(X), there is x ∈ X with x∗(x) = kxk = 1. If we denote by Q the quotient map from X onto X/Y = (Y ⊥)∗, it follows from (2) that Q(x) = e. Hence Q(BX ) ⊇ Ext(BX/Y ) and thus by the Krein–Milman theorem

Q(BX ) = BX/Y which means that Y is proximinal.

∗ Lemma III.7. Let X be a subspace of c0. For any x ∈ SX∗ , we denote J(x∗) = {x∗∗ ∈ X∗∗; x∗∗(x∗) = kx∗∗k = 1}.

If x∗ ∈ NA(X), there exists a w∗-neighbourhood V of x∗ such that if y∗ ∈ ∗ ∗ V ∩ SX∗ , then J(y ) ⊆ J(x ). 8 g. godefroy

∗ ∗ Indeed, let j : X → c0 be the canonical injection, and j : `1 → X be the corresponding quotient map. We work by contradiction: if the conclusion ∗ ∗ ∗ ∗ fails, there is a sequence (xn) ⊆ SX∗ such that w -lim(xn) = x and

∗ ∗ J(xn) ⊂/J(x ) (3)

∗ ∗ ∗ ∗ ∗ ∗ for all n. Pick yn ∈ `1 with kynk = 1 and j (yn) = xn. If we let for z ∈ S`1

∗ ∗ J˜(z ) = {t ∈ `∞; ktk = 1 = t(z )} we formally have when kj∗(z∗)k = 1 ¡ ¢ j∗∗ J(j∗(z∗)) = j∗∗(X∗∗) ∩ J˜(z∗). (4)

∗ ∗ We may and do assume that the sequence (yn) is w -convergent in B`1 to ∗ ∗ ∗ ∗ ∗ ∗ ∗ y . Clearly j (y ) = x , and y (x) = ky k = 1. Hence, y ∈ NA(c0) and it follows that the set

∗ ∗ supp(y ) = {k ≥ 1; y (ek) 6= 0} is ﬁnite. It follows from (3) and (4) that one has

˜ ∗ ˜ ∗ J(yn) ⊂/ J(y ) for all n. On the other hand, it is clear that

˜ ∗ ∗ ∗ ∗ J(y ) = {t ∈ S`∞ ; t(ek) = sign(y (ek)) for all k ∈ supp(y )}

˜ ∗ ∗ ∗ ∗ with a similar formula for J(yn). Now since w -lim(yn) = y , it follows that for n large enough, one has for all k ∈ supp(y∗) that

∗ ∗ sign(yn(k)) = sign(y (k))

˜ ∗ ˜ ∗ and thus J(yn) ⊂ J(y ), which is a contradiction. To conclude the proof of Theorem III.5 it suﬃces by Lemma III.6 to show that Y ⊥ is polyhedral. Since it is contained in NA(X), we may apply Lemma ∗ ⊥ ∗ III.7 to every x ∈ Y . By compactness, we ﬁnd {xi ; i ≤ n} in SY ⊥ and ∗ ∗ w -open neighbourhoods Vi of xi such that [n SY ⊥ ⊆ Vi i=1 the banach space c0 9 and ∗ ∗ J(y ) ⊆ J(xi ) ∗ if y ∈ Vi. Therefore, [ [n ∗ ∗ ∗ {J(y ); y ∈ SY ⊥ } = J(xi ) i=1 it follows that for any t ∈ (Y ⊥)∗,

Remark III.8. By using sharper analytical tools, one can show [19] that under the assumptions of Theorem III.5, the proximinal subspace Y is in fact strongly proximinal, in the following sense. For every x ∈ X, denote d(x, Y ) = inf{kx−yk; y ∈ Y } and PY (x) = {y ∈ Y ; kx−yk = d(x, Y )}. Then for every ε > 0, there is δ > 0 such that if y ∈ Y and kx − yk < d(x, Y ) + δ, 0 0 then there is y ∈ PY (x) such that ky − y k < ε. In other words, if y is almost a nearest point to x, then y is close to a nearest point. Another result of [19] is that under the assumption of Theorem III.5, the multivalued map PY (·) has a continuous selection. The conclusion of Lemma III.7 and the above assertions, are satisﬁed for a collection of spaces which is quite larger that the class of subspaces of c0. However, it provides strong restrictions on the isometric structure. We now provide a proof of a result from [12]. If x∗ ∈ X∗, we denote

∗ ∗ ∗ JX (x ) = {x ∈ SX ; x (x) = kx k}.

We recall that a boundary is a subset B of SX∗ on which every x ∈ X attains its norm. With this notation, the following holds:

Proposition III.9. Let X be a separable Banach space such that for all ∗ ∗ ∗ ∗ x ∈ SX∗ ∩ NA, there is a w -open neighbourhood V∗(x ) of x such that ∗ ∗ ∗ ∗ y ∈ V∗(x ) ∩ SX∗ implies JX (y ) ⊆ JX (x ). Then there is a boundary ∗ B ⊆ SX∗ such that no w -accumulation point of B belongs to SX∗ ∩ NA.

Proof. For x ∈ X, we denote

∗ ∗ JX∗ (x) = {x ∈ SX∗ ; x (x) = kxk}. 10 g. godefroy

For all x ∈ SX , the set

Vx = {y ∈ SX ; JX∗ (y) ∩ JX∗ (x) 6= ∅}

∗ is open in (SX , k · k). Indeed if lim kyn − xk = 0 and xn ∈ JX∗ (yn), we may and do assume that ∗ ∗ ∗ w - lim(xn) = x ∈ JX∗ (x). ∗ ∗ ∗ By our assumption, JX (xn) ⊆ JX (x ) if n ≥ N, hence x (yn) = 1 if n ≥ N, and thus JX∗ (yn) ∩ JX∗ (x) 6= ∅ if n ≥ N. Since (SX , k · k) is a separable metric space, by paracompactness and the Lindel¨ofproperty, there is a sequence (xn) ⊆ SX such that:

(i) SX = ∪n≥1Vxn .

(ii) For all x ∈ SX , there is U 3 x open such that {n; Vxn ∩ U 6= ∅} is ﬁnite.

If for x ∈ SX , we denote

Ex = {n ≥ 1; JX∗ (x) ∩ JX∗ (xn) 6= ∅} then by (i) and (ii), En is a non-empty and ﬁnite set. ∗ For all n, JX∗ (xn) is a w -compact subset of NA ∩ SX∗ , hence there is a ﬁnite subset Fn ⊆ JX∗ (xn) such that [ ∗ ∗ JX∗ (xn) ⊆ {V∗(x ); x ∈ Fn}.

We let [ B = Fn. n≥1 The set B is a boundary. Indeed, pick x ∈ S . There is n such that x ∈ V ; x 0 xn0 pick ∗ ∗ ∗ x ∈ JX (x) ∩ JX (xn0 ). ∗ ∗ ∗ ∗ ∗ There is b ∈ Fn0 ⊆ B such that x ∈ V∗(b ). We have JX (x ) ⊆ JX (b ), hence b∗(x) = 1. ∗ Finally, let (bk) ⊆ B and

∗ ∗ ∗ w - lim bk = x k→∞

∗ ∗ ∗ with x 6= bk for all k. We have to show that x ∈/ SX∗ ∩ NA. the banach space c0 11

∗ If not, let x ∈ SX be such that x ∈ JX∗ (x). For k ≥ K, one has

∗ ∗ JX (bk) ⊆ JX (x ).

∗ ∗ Let (nk) be such that bk ∈ JX (xnk ). We note that for any y ∈ SX [ JX∗ (y) ∩ B ⊆ Fn

n∈Ey and thus JX∗ (y) ∩ B is ﬁnite. It follows that we may and do assume that the points (xnk ) are all distinct. If k ≥ K, we have ∗ ∗ xnk ∈ JX (bk) ⊆ JX (x )

∗ ∗ hence x (xnk ) = x (x) = 1 and thus

∗ ∗ JX (xnk ) ∩ JX (x) 6= ∅.

This means that x ∈ Vnk for all k ≥ K. But this cannot be since Ex is ﬁnite.

It should be mentioned that the converse of Proposition III.9 holds true as well [12]. That is, the existence of such a boundary B implies the existence of ∗ ∗ weak open neighbourhoods V∗(x ) satisfying the assumptions of Proposition III.9. We refer to [19, Examples III.5] for applications of this condition of proximinality results. We note that the conditions of Proposition III.9 are hereditary; every polyhedral isometric predual of `1 satisﬁes them.

IV. The space c0 is minimal among non-reflexive spaces which are smooth enough

In contrast with the previous section, we now consider isomorphic properties, which are however related with the existence of special norms. We will show in particular that c0 is minimal among the spaces whose norm locally depends upon ﬁnitely many coordinates. Let us ﬁrst prove that such spaces are Asplund spaces.

Proposition IV.1. Let X be a separable space whose norm k · k locally depends upon ﬁnitely many coordinates. Then X∗ is separable. 12 g. godefroy

Proof. For every x ∈ SX , there are ε(x) > 0 and a linear operator

n(x) Tx : X −→ (R , k · kx) such that if ky − xk < ε(x), then

kyk = kTx(y)kx

(see the comments after Deﬁnition III.2). By the Lindel¨ofproperty, there is a sequence {xn; n ≥ 1} such that [ ¡ ¢ SX ⊆ B xn, ε(xn) . n≥1

n(x) ∗ ∗ Let now Sn be the unit sphere of the space (R , k · kx) , and Kn = Tx (Sn). It is clear that Kn is norm-compact, hence [ B = Kn n≥1

∗ is a norm-separable subset of SX∗ . For all y ∈ SX , there is x ∈ B such that x∗(y) = 1, hence B is a separable boundary. Now [16, Th. III.3] shows that X∗ is separable.

Remark IV.2. An alternative proof of Proposition IV.1 consists into ob- serving that since k · kx is strongly sub-diﬀerentiable (S.S.D.) for all x ∈ SX by Dini’s theorem, the norm of X is S.S.D. as well. Now, every Banach space which has a S.S.D. norm is an Asplund space.

If a space does not contain c0(N), its supply of compact subsets allows to show a compact variational principle ([8]; see [9, Th. V.2.2]) which we state below.

Theorem IV.3. Let X be a Banach space not containing c0(N). let U 3 0 be a bounded open symmetric subset of X. let f : U → R be a continuous function such that:

(i) f(x) > 0 for all x ∈ ∂U, (ii) f(−x) = f(x), (iii) f(0) ≤ 0, the banach space c0 13 then there is a compact subset K of U, and V 3 0 open such that K +V ⊆ U, and such that if we let

fK (x) = sup{f(x + h); h ∈ K} then fK (0) = 0 and fK (x) > 0 if x ∈ V \{0}.

Proof. By a result of Bessaga–Pelczynski [4], if X 6⊃ c0(N) and (xi)i∈N is such that © ª P [N] sup k εixik;(εi) ∈ {−1, 1} < ∞ P then the series ( xi) is unconditionally convergent, and thus the set © ª P [N] E = εixi;(εi) ∈ {−1, 1} is k · k-relatively compact. Let x0 = 0 ∈ U. If x0, ··· , xn have been chosen, let

nXn o Kn = εixi; εi ∈ {−1, 1} i=0 and © ª En = x ∈ X; for all k ∈ Kn, (x + k) ∈ U and f(x + k) ≤ 0 .

Denote © ª αn = sup kxk; x ∈ En and choose xn+1 ∈ En such that α kx k ≥ n . n+1 2

Note that the construction can be continued since 0 ∈ En for all n. Let now K = ∪Kn. Since K ⊆ U, it is bounded, and Bessaga–Pelczynski’s result shows that K is compact and that lim(αn) = 0. We observe now that f ≤ 0 on K, hence K ∩ ∂U = ∅ and fK (0) ≤ 0. Let V be a neighbourhood of 0 such that K + V ⊆ U. If x ∈ V \{0}, there is n ≥ 1 such that kxk > αn, and thus fK (x) > 0; it also follows that fK (0) = 0. 14 g. godefroy

The unit ball of X 6⊃ c0(N) may fail to contain extreme points; however, it contains “extreme compact sets”. Indeed:

Corollary IV.4. If X does not contain c0, there is a compact subset K of BX such that (K + h) 6⊂ BX for all h 6= 0.

Proof. Let U = {x ∈ X; kxk < 2}. We denote

f(x) = dist(x, BX ) = kxk − 1.

Theorem IV.3 provides us with a compact symmetric set K such that fK (0) = 0 (hence K ⊆ BX ) and fK (h) > 0 (hence (K + h) 6⊂ BX ) if 0 < khk < δ for some δ > 0. Clearly, this set K works.

Note that if K is a compact subset of c0, one has ¡ © ª¢ lim sup |x(i)|; x ∈ K, i ≥ n = 0. n→∞

Hence, c0 equipped with its natural norm does not satisfy the conclusion of Corollary IV.4, which says that c0 is minimal for this property. It follows that it is also minimal for local dependence upon ﬁnitely many coordinates.

Corollary IV.5. If the norm of X locally depends upon ﬁnitely many coordinates and dim X = ∞, then X contains c0.

Proof. If not, pick K ⊆ BK satisfying the conclusion of Corollary IV.4. We may and do assume that 0 ∈/ K. For all x ∈ K, there is ε(x) > 0 such that if kx − yk < ε(x), then

kyk = ϕx(Tx(y)) where Tx is a ﬁnite rank map. By compactness, we have

[n ¡ ¢ K ⊆ B xi, ε(xi) . i=1

n Since dim X = ∞, the space H = ∩i=1 Ker(Txi ) is not reduced to {0}. Hence we can pick h ∈ H with 0 < khk < ε = inf1≤i≤n ε(xi). Clearly, (K +h) ⊆ BX , contradicting our choice of K. the banach space c0 15

The compact variational principle (Theorem IV.3) can be used to provide some uniform continuity in spaces which do not contain c0. For instance, if k X 6⊃ c0, and if there is a C -smooth bump function b0 : X → R (k ≥ 1), then there is a Ck-smooth bump function b such that b(k−1) is uniformly continuous 2 (see [9, Th. V.3.2]). In particular, if X 6⊃ c0 and X has a C -smooth bump, then X is superreflexive of type 2 (see [9, Cor. V.3.3]). ∞ Note that c0 has a C -smooth bump which locally depends upon finitely many coordinates (see [9, Chapter V]). Indeed, pick f : R → R a C∞- smooth function such that f(t) = 1 if |t| ≤ 1 and f(t) = 0 if |t| ≥ 2. If x = (x(i)) ∈ c0, we let Y∞ b(x) = f(x(i)). i=1 Note that for this bump function b, like for any differentiable bump function 0 on c0, the derivable b is not uniformly continuous.

V. The space c0 is maximal among asymptotically flat spaces

The following “modulus of asymptotic smoothness” has been defined in [33]. If x ∈ SX , τ > 0 and Y is a linear subspace of X, define © ª ρ(x, τ, Y ) = sup kx + τyk − 1; y ∈ SY and then © ª ρ(x, τ) = inf ρ(x, τ, Y ); dim X/Y < ∞ and finally © ª ρX (τ) = sup ρ(x, τ); x ∈ SX .

In other words, ρX (τ) measures the uniform smoothness of the norm of X, when one is allowed to “neglect” at every point x ∈ SX a well-chosen ﬁnite dimensional space. Following [29], one says that X is asymptotically uniformly smooth if

ρ (τ) lim X = 0. τ→0 τ

Let us say that X is asymptotically uniformly ﬂat if there is τ0 > 0 such that ρX (τ0) = 0 (or equivalently, ρX (τ) = 0 for all τ ∈ [0, τ0]). It is easily seen that when (X = c0, k · k∞), then ρX (1) = 0. But it turns out that c0 is the largest space with this property. 16 g. godefroy

Theorem V.1. Let X be an asymptotically uniformly ﬂat Banach space. Then X is isomorphic to a subspace of c0.

∗ ∗ ∗ Proof. Let x ∈ SX and x ∈ SX∗ such that x (x) = 1. Let (un) be a ∗ ∗ ∗ ∗ ∗ sequence in X with w -lim(un) = 0 and lim kx + unk = α > 0. Pick any ε > 0. Since X is asymptotically uniformly flat, there is τ0 > 0 independent of x such that we can find Y ⊂ Ker(x∗) finite codimensional, such that for all y ∈ SY , kx + τ0yk ≤ 1 + ετ0. ∗ ∗ Since w -lim(un) = 0, there exists by [20, Lemma 2.5] a sequence (yn) ⊂ SY with w-lim(yn) = 0 such that α limhu∗ , y i ≥ n n 2 and thus we have α limhx∗ + u∗ , x + τ y i ≥ 1 + τ . n 0 n 0 2 It follows that α ∗ ∗ 1 + τ0 2 lim kx + unk ≥ , 1 + τ0ε hence since ε > 0 is arbitrary α lim kx∗ + u∗ k ≥ 1 + τ . (5) n 0 2 ∗ ∗ Summarizing, we have shown (5) for all w -null sequences (un) with ∗ ∗ lim kunk = α, provided that x is norm-attaining. Since such functionals ∗ are norm-dense by the Bishop–Phelps theorem, (5) holds for every x ∈ SX∗ . In the terminology of [21], (5) means that the norm of X is “Lipschitz w∗-Kadec–Klee”. Now [21, Th. 2.4] shows that X is isomorphic to a subspace of c0. It should be noted that the proof of [21, Th. 2.4], which is quite technical, can be made simpler if one assumes that X has a shrinking F.D.D., in which case it boils down to a skipped blocking argument. Now if E is an arbitrary space with E∗ separable, there is Y ⊆ E such that Y and (E/Y ) both have a shrinking F.D.D. To conclude the proof, it suffices to show that “being a subspace of c0” is a 3-space property (see Cor II.3 above). This approach is followed in [29] (see also [17, Th. V.7]). the banach space c0 17

A motivation for Theorem V.1 is the following non-linear result: if (X, k · kX ) is asymptotically uniformly ﬂat and U : X → Y is a bijective map such that U and U −1 are both Lipschitz maps, then the formula

½ ∗ 0 ¾ ∗ |y (Ux − Ux )| 0 2 0 |||y ||| ∗ = sup ;(x, x ) ∈ X , x 6= x Y kx − x0k defines an equivalent dual norm on Y ∗ whose predual norm is asymptotically uniformly flat. It follows from this (non trivial) fact and Theorem V.1 that the class of subspaces of c0 is stable under Lipschitz isomorphisms, and in particular that a space which is Lipschitz isomorphic to c0 is linearly isomorphic to c0 [21]. It should be noted that “asymptotic uniform flatness” is in fact equivalent to “Lipschitz w∗-uniform Kadec Klee”; more generally, asymptotic uniform smoothness is equivalent to w∗-uniform Kadec Klee. This is a non-reflexive version of the well-known duality between uniform convexity and uniform smoothness. We recall that a Banach space Y is said to be M-embedded if for every y∗ ∈ Y ∗ and t ∈ Y ⊥ ⊂ Y ∗∗∗, one has ky∗ + tk = ky∗k + ktk. We refer to [28] for numerous examples and applications. Note that any non reflexive M- embedded space contains an isomorphic copy of c0 [2] (see [23, Th. 3.5] for a more general result). It is clear that (c0, k·k∞) is M-embedded. We prove now that this property 1 characterizes c0 among the isomorphic preduals of ` .

Corollary V.2. Let X be a separable L ∞ space. If X is M-embedded, then X is isomorphic to c0.

Proof. Since X is M-embedded, the weak∗ and weak topologies coincide ∗ ∞ on SX∗ (see [28, Cor. III.2.15]) and thus X is separable. Since X is L , X∗ is a separable L 1 dual space, and thus [31] it is isomorphic to `1. Let us show that k · kX is asymptotically uniformly ﬂat. ∗ ∗ ∗ ∗ To show this, take a sequence (un) with w -lim(un) = 0 and lim kunk = ε, ∗ ∗ ∗ such that lim kx + unk ≤ 1. We have to show that kx k ≤ 1 − Kε for some ﬁxed constant K > 0. Since X∗ is isomorphic to `1, it has the strong Schur property (see [5]). Hence there is a subsequence (u∗ ) of (u∗ ) such that nk n ku∗ − u∗ k ≥ ε/2 for all k 6= l, and the strong Schur property provides a nk nl ∗ further subsequence which we denote (vn) which is (K0ε)-equivalent to the 1 ∗ ∗ ∗∗∗ canonical basis of ` . It follows that (vn) has a w -cluster point G in X 18 g. godefroy such that dist(G, X∗) ≥ Kε for some K > 0. Note that G ∈ X⊥ since w∗- ∗ ∗ ∗ ∗ lim(vn) = 0 and thus dist(G, X ) = kGk ≥ Kε. Now since lim kx + vnk ≤ 1, we have kx∗ + Gk ≤ 1 and thus by the M-ideal property,

kx∗k ≤ 1 − kGk ≤ 1 − Kε.

By Theorem V.1, X is isomorphic to a subspace of c0. The conclusion follows ∞ since any L subspace of c0 is isomorphic to c0. Corollary V.2 has been originally shown in [25] through an application of Zippin’s converse to Sobczyk’s theorem [37]. We refer to [21] for non separable versions of Corollary V.2.

VI. Approximation properties in subspaces of c0

We recall that a Banach space X has the approximation property (in short, A.P.) if for any compact set K and any ε > 0, there is a finite rank operator R such that kx − Rxk < ε for all x ∈ K. If there is M < ∞ such that R can be chosen with kRk ≤ M, we say that X has the bounded approximation property (B.A.P.); if M = 1, X has the metric approximation property (M.A.P.). It is easily seen that a separable space X has the B.A.P. if and only if there is a sequence (Rn) of finite rank operators such that kRnk ≤ M for all n and lim kx − Rn(x)k = 0 ( with M = 1 if and only if X has M.A.P.). A remarkable result from [6] asserts that if X is any separable Banach space with M.A.P., there is a sequence (Rn) of finite rank operators such that

(i) kRnk ≤ 1 for all n.

(ii) lim kx − Rn(x)k = 0 for all x ∈ X.

(iii) RnRk = RkRn for all n, k. We now investigate some aspects of the approximation properties in subspaces of c0.

Theorem VI.1. Let X be an M-embedded separable Banach space, such that there exists a sequence (Rn) of ﬁnite rank operators satisfying (ii), (iii) 0 and such that kRnk < 2 − ε for some ε > 0. Then there is a sequence (Rn) of ﬁnite rank operators satisfying (i), (ii) and (iii). Note that being M-embedded is a hereditary property; hence Theorem VI.1 applies in particular to every subspace of c0. the banach space c0 19

∗ ∗ Proof. We consider the conjugate operators (Rn) on X . It follows from (ii) that ∗ ∗ ∗ ∗ w - lim Rn(x ) = x for all x∗ ∈ X∗. We claim that [ ∗ ∗ ∗ Rn(x ) = X . (6) n≥1 ∗∗ ∗ ∗ ∗ ∗ If not, there is F ∈ X with kF k = 1 such that F (Rn(x )) = 0 for all x ∈ X and for all n. We pick x∗ ∈ X∗ such that kx∗k ≤ (2 − ε)−1 and F (x∗) > 1/2. We have ∗ ∗ ∗ ∗ kRn(x )k ≤ kRnkkx k ≤ 1. ∗∗∗ ∗ ∗∗∗ ∗ ∗ Let t ∈ X be a w -cluster point in X to the sequence (Rn(x )). It is ∗ clear that F (t) = 0 and t|X = x . Since X is M-embedded, we have 1 kx∗ − tk = ktk − kx∗k ≤ 1 − F (x∗) < . 2 On the other hand, F (x∗ − t) > 1/2, a contradiction which shows (6). We now claim that in fact

∗ ∗ ∗ lim kx − Rk(x )k = 0 (7) for all x∗ ∈ X∗. Indeed by (6), for any ε > 0, there is y∗ ∈ X∗ and n ≥ 1 such that ∗ ∗ ∗ kx − Rn(y )k < ε. Using (iii), we have for all k ≥ 1

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ kRk(x ) − RnRk(y )k = kRk(x ) − RkRn(y )k ≤ 2ε. ∗ ∗ Now since Rn is a conjugate ﬁnite rank operator, it is (w -k · k) continuous and thus ∗ ∗ ∗ ∗ ∗ lim kRn(y ) − RnRk(y )k = 0. k It follows that there is A(ε) such that if k ≥ A(ε) then

∗ ∗ ∗ kx − Rk(x )k < 4ε and this shows (7). To conclude the proof, we note that (7) implies that X∗ is a separable dual with the approximation property. By a theorem of Grothendieck (see [32, Th. I.e.15]) it follows that X∗ has the M.A.P., and thus X has M.A.P. as well (see [32, Th. I.e.7]) and we may apply [6]. 20 g. godefroy

We refer to [24] for an alternative construction of commuting approximating sequences in spaces not containing `1, which applies in particular to M-embedded spaces. It follows, immediately from Theorem VI.1 and [6] that we have

Corollary VI.2. Let X be a separable M-embedded space. If there exists a Banach space Y with the metric approximation property such that dBM (X,Y ) < 2 then X has the metric approximation property. It is natural to wonder whether 2 can be replaced by an arbitrary constant in Theorem VI.1 and Corollary VI.2. It is not so, as shown by the following example due to W.B. Johnson and G. Schechtman (see [30]).

Theorem VI.3. There exists a subspace X of c0, such that there is a sequence (Rn) of ﬁnite rank operators satisfying (ii), (iii) and sup kRnk ≤ 8, but X fails the metric approximation property.

Proof. It follows from Enﬂo’s theorem [10] that there is a subspace Y of c0 failing A.P. (see [32, p. 37]). Let (Yn)n≥1 be an increasing sequence of ﬁnite dimensional subspaces of Y such that Y = ∪nYn. We consider the space

c((Yn)) = {yn ∈ Yn :(yn) is norm-convergent}.

When equipped with the supremum norm, c((Yn)) is clearly a Banach space. The map

L : c((Yn)) −→ Y (yn) 7→ lim(yn) is a quotient map, whose kernel is the space c0((Yn)). We observe that by Corollary II.3, the space c((Yn)) is isomorphic to a subspace of c0, since its subspace c0((Yn)) and the corresponding quotient space Y are both isomorphic (in fact, isometric) to subspaces of c0. An examination of the proof of Corollary II.3 shows in the notation used there that kV k ≤ 2 −1 and kV |Im(V )k ≤ 4. It follows that there is a subspace X of c0 such that ¡ ¢ dBM c0((Yn)),X ≤ 8.

Since c0((Yn)) trivially has the M.A.P., the existence of the sequence (Rn) with (ii), (iii) and sup kRnk ≤ 8 follows. the banach space c0 21

The proof of Theorem VI.1 shows in particular that if a subspace of c0 has the M.A.P. then its dual has the M.A.P. as well. Hence it suﬃces to show that the space X∗ fails A.P.; this follows from the

Fact VI.4. The space Y ∗ is isomorphic to a complemented subspace of ∗ c((Yn)) . ∗ ∗ ∗ Indeed, L : Y → c((Yn)) is an isomorphic embedding, and it suffices to ∗ ∗ ∗ find P : c((Yn)) → Y such that PL = IdY ∗ . For all n ≥ 1, let jn : Yn → c((Yn)) be defined as follows: if jn(y) = z then ( 0 if k < n zk = y if k ≥ n.

∗ ∗ ∗ Pick any f ∈ c((Yn)) , and let fn = jn(f). Clearly, kfnk ≤ kfk. Let fn ∈ Y be an extension of fn to Y with kfnk = kfnk. Fix a free ultraﬁlter U on N. ∗ It is easily seen that w -limn→U (fn) does not depend upon the choice of the extension fn; we can therefore set

∗ P (f) = w - lim fn. n→U

∗ ∗ ∗ ∗ ∗ ∗ Finally, PL (y ) = y for every y ∈ Y . Indeed, if (gn) ⊆ Y is any sequence ∗ ∗ ∗ ∗ such that kgnk ≤ ky k and gn = y on Yn, then (gn) is w -convergent to y . To conclude the proof of Theorem VI.3, we observe that since Y fails A.P., ∗ ∗ the space Y fails A.P. as well (see [32, Th. I.e.7]). Hence c((Yn)) fails A.P. since it contains a complemented subspace which fails it. Note that it follows from Theorem VI.1 that the space X, which has the B.A.P., is such that dBM (X,Y ) ≥ 2 for all spaces Y which have the M.A.P. It is a well-known and important open problem [32, Pb. I.e.21] whether every Banach space which has the B.A.P. is isomorphic to a Banach space with the M.A.P.; or equivalently by [6], if every Banach space with the B.A.P. has this property with commuting operators. The space X of Theorem VI.3 is, by its construction, isomorphic to a space which has M.A.P.

VII. Open problems

§ II. As indicated in Remark II.4, Sobczyk’s theorem fails in the nonseparable case. However, the separable complementation property shows that c0 is complemented in every W.C.G. space X containing it. This extend to 22 g. godefroy

c0(Γ) spaces if dens(X) < ℵω0 , but not further [1]. We refer to [9, Th. II.8.3] for a link between extensions of G-smooth norms and the existence of linear projections. Let us mention that the relevant Problem II.2 from [9, p. 90] on extensions of Fr´echet smooth norms is still open. § III. Proximality of ﬁnite codimensional subspaces can be understood as an n-dimensional version of Bishop–Phelps theorem (see Prop. III.4). It is not known whether every Banach space contains a proximinal subspace of codimension 2. It is also an open problem to know if every dual space contains a 2-dimensional subspace consisting of norm-attaining functionals.

§ IV. The gist of this section is that c0 is minimal among non superreﬂexive Ck smooth spaces, with k > 1. A related open problem is whether every continuous convex function f on c0 has points of “Lipschitz smoothness” (see [11]); that is, let f : c0 → R be a convex continuous function. Does there exists a point x0 of Fr´echet smoothness of f such that

0 2 f(x0 + h) = f(x0) + f (x0)h + O(khk )?

Note that Fr´echet smoothness means that the remainder is o(khk). ∞ It follows from the existence of a C smooth bump function on c0 that ∞ every continuous function g : c0 → R is uniformly approximable by C smooth functions (see [9, Th. VIII.3.2]). If g is assumed to be uniformly continuous, it is uniformly approximable by real-analytic functions ([7], [14]). It is not known if this conclusion still holds for continuous functions. It would be interesting to link this topic with Gowers’ stability theorem ([27]; see [3, Th. 13.18]).

§ V. The results from [20] (see [3, Th. 10.17]) show that c0 is somehow a “rigid” space. We refer again to [27] (see [3, Th. 13.18]) for an independent and deep result of non-linear rigidity of c0. It is not known whether a Banach space which is uniformly homeomorphic to c0 is isomorphic to c0, although it can be shown that such a space is an isomorphic predual of `1 which shares many features of c0 [21]. Corollary V.2 is one of the characterizations of c0 1 among preduals of ` . It is not known whether c0 is the only isomorphic predual of `1 which has Pelczynski’s property (u) (see [23, Th. 7.7] for a partial result). § VI. No eﬀort was made in the proof of Theorem VI.3 to tighten the constant and it is unlikely that 8 is the critical value. A natural guess is 2. No example is known of a subspace of c0 having A.P. but failing B.A.P. It is not known whether the assumption of commutativity (iii) can be removed in the banach space c0 23

Theorem VI.1; in other words, if a subspace of c0 with λ-B.A.P. for λ < 2 has the M.A.P. Note that [26, Ex. 4.7] shows that the proof of Theorem VI.1 fails to provide a positive answer. By [22, Prop. 3.2], a subspace X of c0 with the metric approximation property whose dual embeds into L1 is isomorphic to a quotient of c0; it is not known whether one can dispense with assuming the metric approximation property. Acknowledgements These notes originated in a summer course taught in Laredo (Spain) in September 1999. I am glad thank the organizer of this summer School, Professor M. Gonz´alez,for his kind invitation. I am also grateful to Professor J. Moreno who kindly oﬀered to publish these notes in Extracta Mathematicae.

References [1] Argyros, S., Castillo, J., Granero, A., Jimenez,´ M., Moreno, J., On Sobczyk’s theorem, preprint. [2] Behrends, E., Harmand, P., Banach spaces which are proper M-ideals, Studia Math., 81 (1985), 159 – 169. [3] Benyamini, Y., Lindenstrauss, J., “Geometric Nonlinear Functional Analysis, Vol. 1”, A.M.S. Colloquium Publications 48, American Mathemat- ical Society, Providence, RI, 2000. [4] Bessaga, C., Pelczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math., 17 (1958), 151 – 164. [5] Bourgain, J., Rosenthal, H.P., Martingales valued in certain subspaces of L1, Israel J. Math., 37 (1980), 54 – 75. [6] Casazza, P., Kalton, N.J., Notes on approximation properties in separable Banach spaces, in “Geometry of Banach Spaces (Strobl, 1989)”, P.F.X. Müllerand W. Schachermayer (eds.), London Math. Soc. Lecture Notes Se- ries 158, Cambridge Univ. Press, Cambridge, 1990, 49 – 63. [7] Cepedello, M., Hajek, P., Analytic approximations of uniformly continuous operators, J. Math. Anal. App., to appear. [8] Deville, R., Fabian, M., Principles variationels et diffeéntiabilitéd’ applications définiessur un espace de Banach, Publ. Math. Besan¸con,Analyse non linéaire, 10 (1989), 79 – 102. [9] Deville, R., Godefroy, G., Zizler, V., “Smootness and Renormings in Banach Spaces”, Pitman Monographs 64, Longman Ed., Harlow, 1993. [10] Enflo, P., A counterexample to the approximation property in Banach spaces, Acta Math., 130 (1973), 309 – 317. [11] Fabian, M., Lipschitz-smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc., 51 (1985), 113 – 126. [12] Fonf, V., Lindenstrauss, J., On quotients of polyhedral spaces, preprint. [13] Fonf, V., Lindenstrauss, J., Phelps, R.R., Infinite dimensional convexity, in “Handbook on Banach Spaces”, Elsevier Ed., to appear. 24 g. godefroy

[14] Fry, R., Analytic approximation on c0, J. Func. Anal., 158 (1998), 509 – 520. [15] Garkavi, A.L., Helly’s problem and best approximation in spaces of continuous functions, Izv. Akad. Nauk. SSSR, Ser. Math., 31 (1967), 641 – 656. [16] Godefroy, G., Boundaries of a convex set and interpolation sets, Math. Ann., 277 (1987), 173 – 184. [17] Godefroy, G., Renormings of Banach spaces, in “Handbook on Banach Spaces”, Elsevier Ed., to appear. [18] Godefroy, G., Indumathi, V., Proximality in subspaces of c0, J. Approx. Theory, 101 (1999), 175 – 181. [19] Godefroy, G., Indumathi, V., Strong proximinality and polyhedral spaces, Revista Matemática, to appear. [20] Godefroy, G., Kalton, N.J., Lancien, G., Subspaces of c0(N) and Lipschitz isomorphisms, Geom. Funct. Anal., 10 (2000), 798 – 820. [21] Godefroy, G., Kalton, N.J., Lancien, G., Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc., to appear [22] Godefroy, G., Kalton, N.J., Li, D., On subspaces of L1 which embed into `1, J. Reine Angew. Math., 471 (1996), 43 – 75. [23] Godefroy, G., Kalton, N.J., Saphar, P.D., Unconditional ideals in Banach spaces, Studia Math., 104 (1993), 13 – 59. [24] Godefroy, G., Kalton, N.J., Approximating sequences and bidual projections, Quart. J. Math. Oxford (2), 48 (1997), 179 – 202. [25] Godefroy, G., Li, D., Some natural families of M-ideals, Math. Scand., 66 (1990), 249 – 263. [26] Godefroy, G., Saphar, P.D., Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math., 32 (1988), 672 – 695. [27] Gowers, W.T., Lipschitz functions on classical spaces, European J. Combin., 13 (1992), 141 – 151. [28] Harmand, P., Werner, D., Werner, W., M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer-Verlag, Berlin, 1993. [29] Johnson, W.B., Lindenstrauss, J., Preiss, D., Schechtman, G., Almost Fréchet differentiability of Lipschitz mapping between infinite dimensional Banach spaces, to appear. [30] Johnson, W.B., Oikhberg, T., Separable lifting properties and extensions of local reflexivity, Illinois J. Math., to appear. [31] Lewis, D.R., Stegall, C., Banach spaces whose duals are isomorphic to `1(Γ), J. Func. Anal., 12 (1973), 177 – 187. [32] Lindenstrauss, J., Tzafriri, L., “Classical Banach Spaces. I. Sequence Spaces.”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92., Springer-Verlag, Berlin-New York, 1977. [33] Milman, V.D., Geometric theory of Banach spaces. II. Geometry of the unit ball (Russian), Uspehi Mat. Nauk, 26 (1971), 6(162), 73 – 149. English trans- lation: Russian Math. Surveys, 26 (1971), 6, 71 – 163. [34] Phelps, R.R., Chebyshev subspaces of finite codimension in C(X), Pacific J. Math., 13 (1963), 647 – 655. [35] Sobczyk, A., Proyection of the space m on its subspace c0, Bull Amer. Math, Soc., 47 (1941), 938 – 947. the banach space c0 25

[36] Veech, W.A., Short proof of Sobczyk’s theorem, Proc. Amer. Math. Soc., 28 (1971), 627 – 628. [37] Zippin. M., The separable extension problem, Israel J. Math., 26 (1977), 372 – 387.